Time-frequency analysis involves monitoring the changes in the frequency spectrum of a system over time. The analysis is of importance in a wide variety of fields of science and engineering. The old work-horse of frequency analysis, the fast Fourier transform (“FFT”), can be utilized in time-frequency analysis by segmenting data into time windows. By monitoring changes in the power spectrum in each time window, the presence or absence of activity at certain frequencies within that time window can be determined. The segmentation process, however, introduces artifacts and precludes finer time resolution within the time windows. To address this problem, several refinements of the FFT approach have been proposed. These refinements, however, are all somewhat labor intensive, requiring judgments to be made about time window sizes and windowing functions, which may be different for different frequency ranges.
Over the last twenty years, wavelet transforms have become an important new tool for performing time-frequency analysis. A wavelet includes an oscillatory waveform that has a fairly well-defined frequency, and which exists only for a brief period of time. By convolving time series data with a suitably chosen wavelet, a determination of whether an oscillation of a certain frequency is present at a certain interval in time can be made, and in a manner that is more convenient and less susceptible to artifact than time windowed FFT. There are many kinds of wavelets and many applications of wavelet transforms to electroencephalography (“EEG”) analysis.
A defining property of wavelets is the admissibility criterion, a consequence of which is that the mean of the wavelet when averaged over all time must equal zero. This criterion ensures that a stable inverse transform exists. A stable inverse transform is important for reliable signal transmission and reconstruction. However, in many fields of science and engineering, signal reconstruction is not the desired end result; rather, it may only be desirable to detect whether oscillations of certain frequencies appear, at what times they appear, and for what duration of time they exist. For example, in the brain, it is known that oscillations in the theta (4-8 Hz) and gamma (30-100 Hz) ranges are associated with cognitive activity. Oscillations from these two frequency ranges are sometimes phase-coupled, such that the faster gamma frequencies ride entirely on the crests, or troughs, of slower theta rhythms. Of increasing clinical interest are high frequency oscillations (“HFOs”) in the range of 200-500 Hz. These oscillations tend to occur more frequently in brain regions that are epileptogenic, and so may be useful as a marker for tissue that should be surgically resected in patients with refractory epilepsy. For these applications, it might be desirable to relax the admissibility criterion. Relaxing the admissibility criterion results in a wide class of waveforms called pseudo-wavelets, or quasi-wavelets, which have found application in the study of turbulence and other complex phenomena.
One such pseudo-wavelet was previously described for discretized data applications by D. Hsu, et al., in “An Algorithm for Detecting Oscillatory Behavior in Discretized Data: the Damped-Oscillator Oscillator Detector,” 2007; arXiv:0708.1341v1 [q-bio.QM]. The pseudo-wavelet discussed by Hsu, et al., corresponded to using a mathematical model of a frictionally damped harmonic oscillator to detect data oscillations of the same frequency as the damped harmonic oscillator. While this pseudo-wavelet included a friction factor to damp the mathematical oscillator, the friction factor was used as a free parameter to control noise in the time-frequency analysis process. Thus, while the pseudo-wavelet discussed in Hsu, et al., provided temporal resolution benefits, it was still constrained by existing spectral resolution limitations.
It would therefore be desirable to provide a system and method for time-frequency analysis in which higher temporal resolution than that achievable with Fourier analysis is achievable, while simultaneously increasing the achievable spectral resolution of the analysis. Preferably, such a dual increase in temporal and spectral resolution would be provided without an undue increase in computational burden.